High-Order Breather Solutions, Lump Solutions, and Hybrid Solutions of a Reduced Generalized (3 + 1)-Dimensional Shallow Water Wave Equation

Jing Wang,Biao Li

doi:10.1155/2020/9052457

Abstract

We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived. By using the resulting N-soliton expression and reasonable constraining parameters, we concisely construct the high-order breather solutions, which have periodicity in x,y-plane. By taking a long-wave limit of the breather solutions, we have obtained the high-order lump solutions and derived the moving path of lumps. Moreover, we provide the hybrid solutions which mean different types of combinations in lump(s) and line wave. In order to better understand these solutions, the dynamic phenomena of the above breather solutions, lump solutions, and hybrid solutions are demonstrated by some figures.

Highlights

Bilinear FormWe start from a potential field q to construct the bilinear form of the (3 + 1)-dimensional shallow water wave equation, which is defined by u(x, y, t) c(t)qxx,
Is equation has been used in weather simulations, tidal waves, river and irrigation, and tsunami prediction and researched in different ways
We start from a potential field q to construct the bilinear form of the (3 + 1)-dimensional shallow water wave equation, which is defined by u(x, y, t) c(t)qxx, (2)

Summary

Bilinear Form

We start from a potential field q to construct the bilinear form of the (3 + 1)-dimensional shallow water wave equation, which is defined by u(x, y, t) c(t)qxx,. Where c c(t) is a function to be determined later and q q(x, y, z, t). Substituting the transformation equation (2) into equation (1) and integrating equation(1) with respect to x twice, one can obtain the following equation: E(q) c(t)qyt − c(t)qxxxy − 3c(t)2qxxqy (3). E above expression leads to the following bilinear equation:. With the aid of following transformation q 2ln(f)⇔u c(t)qx 2ln(f)xx. Take z x, and equation (5) becomes the following form: B(f · f) 􏼐DyDt − D3xDy + D2x􏼑(f · f)

High-Order Breather and Lump Solutions

First-Order Breather and Lump Solutions

Second-Order Breather and Lump Solutions

Third-Order Breather and Lump Solutions

Hybrid solutions